Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
338 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C71 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C71 |
| prime radical | C23×C71 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ C22×C4×C71 ◁ C2×#97×C71 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 34. M32×C71 | Q8 | semidirect |
| 44. #106×C71 | Q8 | semidirect |
| 116. #13454 | C71 | direct (N × H) |
| 127. C71 | #13454 | direct (N × H) |
| 156. M32 | Q8×C71 | semidirect |
| 166. #106 | Q8×C71 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C71×#3162 | 2 |
| C71×#2159 | 2 |
| C71×#1944 | 2 |
| C71×#1386 | 2 |
| C71×#1377 | 2 |
| #13454 | 71 |